Topics in Complex Analysis

In this course, I plan to cover some topics in Complex Analysis that are usually considered to be beyond the standard second course in complex analysis but are still accessable for a student who have just take a basic course equivalent to SF1628 at KTH. There will be three different parts of the course.

1. Background. We will cover the Riemann mapping theorem and the Dirichlet problem both classically and with more modern approaches. Central topics are the methods of extremal length by Ahlfors and Beurling. We will also cover Montel’s theorem and applications: the little and big Picard theorem. Possibly we will cover the Koebe/Thurston approach to the Riemann mapping theorem by circle packings.
2. Complex dynamics. We will conver some aspects of the theory of complex dynamics of Fatou and Julia from the former turn of the century, based on Montel’s theorem. Possibly we will cover some of the more modern aspects due to Sullivan, Douady, Herman, Yoccoz, McMullen, Shishikura and others based on quasiconformal methods (Ahlfors, Bers and others).
3. Introduction to the theory of Schramm-Loewner Evolutions (SLE). For a long time physisist have been looking for a theory of random curves in some sense similar to Brownian motions, but which are non-selfintersecting. Finally Oded Schramm made such a construction in 2000 based on basis work by Charles Loewner, and this lead to a intese development of a theory that relates complex analysis and probability theorem. There has been an intensive development and two Field’s medals have been awarded in the area,Wendelin Werner, 2006, Stanislav Smirnov, 2010). The basic ideas in the theory are not that difficult and we will be able to give an introduction. For a first overview see wikipedia article .

Suggested reading material and lecture notes will be handed out during the course.  

The exact times for the course will be decided together with the participant.